Math: Pascal's Triangle and the Binomial Theorem Worksheet

Question 1

Write rows 0 through 5 of Pascal's Triangle. Remember that row 0 is the single number 1.

Accepted Answer

Rows 0 through 5 are: row 0 is 1; row 1 is 1, 1; row 2 is 1, 2, 1; row 3 is 1, 3, 3, 1; row 4 is 1, 4, 6, 4, 1; row 5 is 1, 5, 10, 10, 5, 1.

Question 2

Use Pascal's Triangle to expand (x + y)^4.

Accepted Answer

The expansion is x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The coefficients come from row 4 of Pascal's Triangle.

Question 3

Use Pascal's Triangle to expand (a + b)^5.

Accepted Answer

The expansion is a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5. The coefficients come from row 5 of Pascal's Triangle.

Question 4

Expand (2x + 3)^3 completely.

Accepted Answer

The expansion is 8x^3 + 36x^2 + 54x + 27. This comes from (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + 3^3.

Question 5

Find the coefficient of x^3 in the expansion of (x + 2)^5.

Accepted Answer

The coefficient of x^3 is 40. The x^3 term is C(5, 2)x^3(2)^2, which equals 10 times 4 times x^3, so the coefficient is 40.

Question 6

Use the Binomial Theorem to find the fourth term of (3x - 2)^6 when written in descending powers of x.

Accepted Answer

The fourth term is -4320x^3. The fourth term has k = 3, so it is C(6, 3)(3x)^3(-2)^3 = 20 times 27x^3 times -8 = -4320x^3.

Question 7

Find the missing entries in this row of Pascal's Triangle: 1, 7, __, __, __, __, 7, 1.

Accepted Answer

The complete row is 1, 7, 21, 35, 35, 21, 7, 1. Each interior number can be found by adding the two numbers above it, or by using combinations from row 7.

Question 8

Explain why the coefficients in the expansion of (x + y)^n add up to 2^n.

Accepted Answer

The coefficients add up to 2^n because substituting x = 1 and y = 1 gives (1 + 1)^n = 2^n. The expansion then becomes the sum of all the coefficients.

Question 9

Find the sum of the coefficients in the expansion of (2x - 5)^8.

Accepted Answer

The sum of the coefficients is 6561. To find it, substitute x = 1, giving (2(1) - 5)^8 = (-3)^8 = 6561.

Question 10

Find the coefficient of x^4 in the expansion of (x - 3)^7.

Accepted Answer

The coefficient of x^4 is -945. The x^4 term occurs when k = 3, so the term is C(7, 3)x^4(-3)^3 = 35 times -27 times x^4 = -945x^4.

Question 11

A student claims that the middle number in row 8 of Pascal's Triangle is 56. Check the claim and explain whether it is correct.

Accepted Answer

The claim is not correct. Row 8 is 1, 8, 28, 56, 70, 56, 28, 8, 1, so the middle number is 70.

Question 12

Use the Binomial Theorem to write the general term of (a + b)^n, then use it to identify the term containing a^2b^5 in (a + b)^7.

Accepted Answer

The general term is C(n, k)a^(n - k)b^k. In (a + b)^7, the term containing a^2b^5 has k = 5, so it is C(7, 5)a^2b^5 = 21a^2b^5.

Math: Pascal's Triangle and the Binomial Theorem

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